Asymptotic behavior of Linear and Integer Programming and the stability of the Castelnuovo-Mumford regularity

In this talk I will explain a connection between Commutative Algebra and Linear and Integer Programming. In the first part, it is explained how one can translate the problem of bounding the index of stability of the Castelnuovo-Mumford regularities of the integral closures of powers of a monomial ideal into an Integer Linear Programming. The second part is devoted to the asymptotic behavior of Linear and Integer Programming with a fixed cost linear functional and the constraint sets consisting of a finite system of linear equations or inequalities with integral coefficients depending linearly on $n$. It is shown that the optima of such Linear Programming problems are a linear function of $n$, while the optima of the corresponding Integer Programming problems are a quasi-linear function of $n$, provided $n\gg 0$. In the last part I give bounds on the indices of stability of the Castelnuovo-Mumford regularities of the integral closures of powers of a monomial ideal and that of symbolic powers of a square-free monomial ideal.